# Volume Integral

1. Sep 19, 2015

### Zondrina

Hi everyone.

I've been curious about a particular symbol, but I've never seen it used or mentioned in any context. I don't really have much information about its usage, so I thought I would ask around and see if anyone knew about its application.

I saw this symbol in Microsoft word.

How do we interpret it, and how do we use it?

I'm familiar with closed surface integrals with differential elements $d \vec S$. We use those when we want to calculate the flux of a field $\vec F$. I'm also familiar with closed surface integrals with differential elements $dS$. We use those when we want to calculate surface area.

What about the closed volume integral above though?

I know we should probably use a differential element $dV$ for a closed volume, and the answer would represent the volume of the object. Is there such thing as a differential volume element $d \vec V$ such that we can extend theorems to the fourth dimension (theorem's like Stoke's theorem and the Divergence theorem)?

2. Sep 19, 2015

### Staff: Mentor

I think the dash box is simply for inserting your integrand.

Its up to you to remember the dV part.

There is a seldom used math symbol called the delambertian thats used in relativity that is the 4D version of the del operator but this isnt it.

3. Sep 19, 2015

### Staff: Mentor

Zondrina, I think you're asking about the integration symbol, not the box to the right. According to this page, https://en.wikipedia.org/wiki/Integral_symbol, that's a closed volume integral. I don't know much more about it, and a quick web search didn't turn up much.

4. Sep 19, 2015

### Zondrina

So the only reason the loop is around the triple integral is to signify the volume is closed.

Does that mean something like the divergence theorem can be written like so:

For a closed volume $V$ such as $x^2 + y^2 + z^2 \leq 1$.

For a volume $V$ that isn't closed such as $x^2 + y^2 + z^2 < 1$, would the theorem would take the form:

Otherwise I don't see any reason to ever have to use the symbol mentioned in the OP.

Last edited: Sep 19, 2015
5. Sep 20, 2015

### leright

I don't know what is meant by "closed volume". What is the difference between a closed volume and an open volume?

6. Sep 20, 2015

### HallsofIvy

Staff Emeritus
In three dimensions there is no such thing as a "closed volume". There can be in higher dimensions, of course.